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SUMMARY:On the behavior of posterior probabilities with additional data: m
onotonicity and nonmonotonicity\, asymptotic rates\, log-concavity\, and T
urán’s inequality - Yosi Rinott (The Hebrew University of jerusalem)
DTSTART;VALUE=DATE-TIME:20240529T110000
DTEND;VALUE=DATE-TIME:20240529T120000
UID:https://new.talks.ox.ac.uk/talks/id/2cabb975-8148-4d9d-bd64-bb0e0d6ef9
ea/
DESCRIPTION:Bayesian statisticians quantify their belief that the true par
ameter is ϑ0 by its posterior probability. The starting question of this
paper is whether the posterior at ϑ0 increases when the data are generate
d under ϑ0\, and how it behaves when the data come from ϑ ≠ ϑ0. Can i
t decrease and then increase\, and thus additional data may mislead Bayesi
an statisticians?\n\nFor data arriving sequentially\, we consider monotoni
city properties of the posterior probabilities as a function of the sample
size with respect to certain stochastic orders\, specifically starting wi
th likelihood ratio dominance.\nWhen the data is generated by ϑ ≠ ϑ0 \
, Doob's consistency theorem says that the posterior at ϑ0 converges a.s.
to zero and therefore its expectation converges to zero. We obtain precis
e asymptotic rates of the latter convergence for observations from an expo
nential family and show that the expectation of the ϑ0 -posterior under
ϑ ≠ ϑ0 is eventually strictly decreasing. Finally\, we show that in a
number of interesting cases this expectation is a log-concave function of
the sample size\, and thus unimodal. In the Bernoulli case we obtain this
result by developing an inequality that is related to Turán’s inequalit
y for Legendre polynomials.\nSpeakers:\nYosi Rinott (The Hebrew University
of jerusalem)
LOCATION:Venue to be announced
TZID:Europe/London
URL:https://new.talks.ox.ac.uk/talks/id/2cabb975-8148-4d9d-bd64-bb0e0d6ef9
ea/
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DESCRIPTION:Talk:On the behavior of posterior probabilities with additiona
l data: monotonicity and nonmonotonicity\, asymptotic rates\, log-concavit
y\, and Turán’s inequality - Yosi Rinott (The Hebrew University of jeru
salem)
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